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Properties

Label 2-950-95.49-c1-0-3
Degree 22
Conductor 950950
Sign 0.466−0.884i0.466 - 0.884i
Analytic cond. 7.585787.58578
Root an. cond. 2.754232.75423
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + 4i·7-s − 0.999i·8-s + (−1 − 1.73i)9-s + 3·11-s − 0.999i·12-s + (1.73 − i)13-s + (2 − 3.46i)14-s + (−0.5 + 0.866i)16-s + (−5.19 − 3i)17-s + 2i·18-s + (3.5 + 2.59i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.204 + 0.353i)6-s + 1.51i·7-s − 0.353i·8-s + (−0.333 − 0.577i)9-s + 0.904·11-s − 0.288i·12-s + (0.480 − 0.277i)13-s + (0.534 − 0.925i)14-s + (−0.125 + 0.216i)16-s + (−1.26 − 0.727i)17-s + 0.471i·18-s + (0.802 + 0.596i)19-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=((0.466−0.884i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=((0.466−0.884i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: 0.466−0.884i0.466 - 0.884i
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ950(49,⋅)\chi_{950} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), 0.466−0.884i)(2,\ 950,\ (\ :1/2),\ 0.466 - 0.884i)

Particular Values

L(1)L(1) ≈\approx 0.634675+0.382663i0.634675 + 0.382663i
L(12)L(\frac12) ≈\approx 0.634675+0.382663i0.634675 + 0.382663i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
5 1 1
19 1+(−3.5−2.59i)T 1 + (-3.5 - 2.59i)T
good3 1+(0.866+0.5i)T+(1.5+2.59i)T2 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2}
7 1−4iT−7T2 1 - 4iT - 7T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1+(−1.73+i)T+(6.5−11.2i)T2 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2}
17 1+(5.19+3i)T+(8.5+14.7i)T2 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2}
23 1+(5.19−3i)T+(11.5−19.9i)T2 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2}
29 1+(−14.5+25.1i)T2 1 + (-14.5 + 25.1i)T^{2}
31 1−2T+31T2 1 - 2T + 31T^{2}
37 1−10iT−37T2 1 - 10iT - 37T^{2}
41 1+(4.5−7.79i)T+(−20.5−35.5i)T2 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.46−2i)T+(21.5+37.2i)T2 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2}
47 1+(23.5−40.7i)T2 1 + (23.5 - 40.7i)T^{2}
53 1+(−5.19+3i)T+(26.5−45.8i)T2 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2}
59 1+(4.5−7.79i)T+(−29.5−51.0i)T2 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2−3.46i)T+(−30.5+52.8i)T2 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.06+3.5i)T+(33.5−58.0i)T2 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2}
71 1+(−3+5.19i)T+(−35.5−61.4i)T2 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2}
73 1+(−0.866−0.5i)T+(36.5+63.2i)T2 1 + (-0.866 - 0.5i)T + (36.5 + 63.2i)T^{2}
79 1+(2−3.46i)T+(−39.5−68.4i)T2 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2}
83 1−3iT−83T2 1 - 3iT - 83T^{2}
89 1+(−3−5.19i)T+(−44.5+77.0i)T2 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2}
97 1+(−14.7−8.5i)T+(48.5+84.0i)T2 1 + (-14.7 - 8.5i)T + (48.5 + 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.02153335175736849314243994800, −9.253883584991822195300021768731, −8.745830232621326750504714274519, −7.86286925961538789347959929882, −6.56491162875177472362685968503, −6.13753774016493014430685776534, −5.10107888811314782736796660770, −3.64398279032656960535210406056, −2.58504170256476707877433006267, −1.29940838718436080897476112136, 0.49204898558125110267532481645, 1.99123612801223462201687630043, 3.85084854726474291964734602701, 4.51059847412809537770235779511, 5.74695227921424646434916344299, 6.62718342087125313502679838885, 7.27331410165023910476046279217, 8.258041408658967622893955828608, 9.036013554772395989147657355060, 9.999938917270649970287659278206

Graph of the ZZ-function along the critical line